Application of He’s homotopy perturbation method to nonlinear shock damper dynamics

Abstract

In order to obtain the equations of motion of vibratory systems, we will need a mathematical description of the forces and moments involved, as function of displacement or velocity, solution of vibration models to predict system behavior requires solution of differential equations, the differential equations based on linear model of the forces and moments are much easier to solve than the ones based on nonlinear models, but sometimes a nonlinear model is unavoidable, this is the case when a system is designed with nonlinear spring and nonlinear damping. Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained; homotopy perturbation method (HPM) is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic; the enhanced HPM is used to solve the nonlinear shock absorber and spring equations.

Analytical Approach to Investigation of Deflection of Circular Plate Under Uniform Load by Homotopy Perturbation Method

“…Basic idea of homotopy-perturbation method mentioned in [2][3][4][5][6][7][8][9][10][11][12].Consider equation (1) (2), according to the homotopy perturbation explained in [2][3][4][5][6][7][8][9][10][11][12], we constructed the following homotopy terms: …”

“…Homotopy perturbation method has been used by many mathematicians and engineers to solve various functional equations. This method was further developed and improved by He and applied to many nonlinear problems [2][3][4][5][6][7][8][9][10][11][12]. In this paper, we consider a solid circular plate with radius a under axisymmetric load, , and set the origin coordinate system at center of plate.…”

Nonlinearities distribution Laplace transform-homotopy perturbation method

“…As it is widely known, the importance of research on nonlinear differential equations is that many phenomena, practical or theoretical, are of nonlinear nature. In recent years, several methods focused to find approximate solutions to nonlinear differential equations, as an alternative to classical methods, have been reported, such those based on: variational approaches (Assas 2007; He 2007; Kazemnia et al 2008; Noorzad et al 2008), tanh method (Evans & Raslan 2005), exp-function (Xu 2007; Mahmoudi et al 2008), Adomian’s Decomposition Method (ADM) (Adomian 1988; Babolian & Biazar 2002; Kooch & Abadyan 2012; Kooch & Abadyan 2011; Vanani et al 2011; Chowdhury 2011; Elias et al 2000), parameter expansion (Zhang & Xu 2007), HPM (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Filobello-Nino et al 2013; Aminikhah 2011; Khan et al 2011; Marinca & Herisanu 2011; He 1998; He 1999; He 2006a; Vazquez-Leal et al 2014; Belendez et al 2009; He 2000; El-Shaed 2005; He 2006b; Vazquez-Leal et al 2012a; Ganji et al 2008; Fereidon et al 2010; Sharma & Methi 2011; Biazar & Ghanbari 2012; Biazar & Eslami 2012; Araghi & Sotoodeh 2012; Araghi & Rezapour 2011; Bayat et al 2014; Bayat et al 2013; Vazquez-Leal et al 2012b; Vazquez-Leal et al 2012c; Filobello-Niño et al 2012; Biazar & Aminikhan 2009; Biazar & Ghazvini 2009; Filobello-Nino et al 2012; Khan & Qingbiao 2011; Madani et al 2011; Ji Huan 2006; Feng et al 2007; Mirmoradia et al 2009; Vazquez-Leal et al 2012; Vazquez-Leal et al 2013), Homotopy Analysis Method (HAM) (Rashidi et al 2012a; Rashidi et al 2012b; Patel et al 2012; Hassana & El-Tawil 2011), and perturbation method (Filobello-Nino et al 2013; Holmes 1995; Filobello-Niño et al 2013b; Filobello-Nino et al 2014) among many others. Also, a ...…”

The Study of Heat Transfer Phenomena Using PM for Approximate Solution with Dirichlet and Mixed Boundary Conditions

“…The assumption that the nonlinear part is small compared to the linear is considered as a disadvantage of the method. There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on: variational approaches [5][6][7]29], tanh method [8], exp-function [9, 10], Adomian's decomposition method [11][12][13][14][15][16], parameter expansion [17], homotopy perturbation method [3,4,16,[18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][34][35][36]38,42], homotopy analysis method [30], Homotopy asymptotic method [1], and perturbation method [39,41] among many others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally [40].…”